Answer: The maximum number of tickets of each kind that can be given away without exceeding $300 is 10 tickets of each kind. Explanation: This problem involves setting up and solving a system of inequalities based on the given constraints: the total cost and the minimum number of tickets of each type. The key concepts involved are linear inequalities, systems of inequalities, and graphical solution methods for linear programming problems. Steps: Define variables: Let $ x $ = number of $10 tickets Let $ y $ = number of $20 tickets Write the constraints: Cost constraint: Total cost cannot exceed $300: \[ 10x + 20y \leq 300 \] Minimum tickets constraint: At least 2 tickets of each kind: \[ x \geq 2 \] \[ y \geq 2 \] Simplify the cost inequality: Divide the entire inequality by 10: \[ x + 2y \leq 30 \] Graphical interpretation: Plot the line $ x + 2y = 30 $ on the coordinate plane. The feasible region is the area below or on this line, and to the right of the lines $ x=2 $ and $ y=2 $. Find the maximum number of tickets: Since the goal is to find the maximum number of tickets of each kind without exceeding $300, and considering the constraints $ x \geq 2 $, $ y \geq 2 $, the maximum occurs at the corner point of the feasible region. Determine the corner points: When $ y=2 $: \[ x + 2(2) = 30 \Rightarrow x + 4 = 30 \Rightarrow x=26 \] Check if $ x \geq 2 $: yes, so point $(26, 2)$. When $ x=2 $: \[ 2 + 2y = 30 \Rightarrow 2y=28 \Rightarrow y=14 \] Check if $ y \geq 2 $: yes, so point $(2, 14)$. Intersection with axes (if applicable): $ x=0 $: \[ 0 + 2y \leq 30 \Rightarrow y \leq 15 \] But $ x \geq 2 $, so ignore $ x=0 $. $ y=0 $: \[ x + 0 \leq 30 \Rightarrow x \leq 30 \] But $ y \geq 2 $, so ignore $ y=0 $. Determine the maximum total tickets: At point $(26, 2)$: total tickets = $ 26 + 2 = 28 $. At point $(2, 14)$: total tickets = $ 2 + 14 = 16 $. The maximum total tickets are at $(26, 2)$, with 28 tickets. Summary: The maximum number of tickets of each kind that can be given away without exceeding $300 is **26 tickets of $10 and 2 tickets of $20**. The total number of tickets given away in this case is 28. Note: The question asks for how many of each kind can be given away, so the answer is 26 tickets of $10 and 2 tickets of $20.

What is the number who chose Italian dressing?

Answer: The number of customers who chose Italian dressing is 90. Explanation: This problem involves understanding the distribution of salad dressing choices among customers, which is represented by a circle graph (pie chart). The key concepts involved are percentage calculations and proportions. The problem asks to find the difference in the number of customers choosing Italian dressing and vinaigrette dressing, given the total number of customers and the percentage distribution. The method involves: Calculating the number of customers for each dressing based on the percentage. Finding the difference between the two numbers. Steps: Identify the total number of customers: Total customers = 200 Determine the percentage for Italian dressing: From the table, Italian dressing = 45% Calculate the number of customers who chose Italian dressing: \[ \text{Number of Italian dressing customers} = \frac{45}{100} \times 200 = 0.45 \times 200 = 90 \] Determine the percentage for Vinaigrette dressing: Vinaigrette = 30% Calculate the number of customers who chose Vinaigrette dressing: \[ \frac{30}{100} \times 200 = 0.30 \times 200 = 60 \] Find the difference between Italian and Vinaigrette dressing customers: \[ 90 - 60 = 30 \] Question asks for the difference between the number who chose Italian dressing and the number who chose vinaigrette dressing: The difference is 30. Note: The question also mentions the number who chose vinaigrette dressing, but the main calculation is to find the number for Italian dressing, which is 90.

Answer: The maximum number of tickets of each kind that can be given away without exceeding $300 is 10 tickets of each kind. Explanation: This problem involves setting up and solving a system of inequalities based on the given constraints: the total cost and the minimum number of tickets of each type. The key concepts involved are linear inequalities, systems of inequalities, and graphical solution methods for linear programming problems. Steps: Define variables: Let $ x $ = number of $10 tickets Let $ y $ = number of $20 tickets Write the constraints: Cost constraint: Total cost cannot exceed $300: \[ 10x + 20y \leq 300 \] Minimum tickets constraint: At least 2 tickets of each kind: \[ x \geq 2 \] \[ y \geq 2 \] Simplify the cost inequality: Divide the entire inequality by 10: \[ x + 2y \leq 30 \] Graphical interpretation: Plot the line $ x + 2y = 30 $ on the coordinate plane. The feasible region is the area below or on this line, and to the right of the lines $ x=2 $ and $ y=2 $. Find the maximum number of tickets: Since the goal is to find the maximum number of tickets of each kind without exceeding $300, and considering the constraints $ x \geq 2 $, $ y \geq 2 $, the maximum occurs at the corner point of the feasible region. Determine the corner points: When $ y=2 $: \[ x + 2(2) = 30 \Rightarrow x + 4 = 30 \Rightarrow x=26 \] Check if $ x \geq 2 $: yes, so point $(26, 2)$. When $ x=2 $: \[ 2 + 2y = 30 \Rightarrow 2y=28 \Rightarrow y=14 \] Check if $ y \geq 2 $: yes, so point $(2, 14)$. Intersection with axes (if applicable): $ x=0 $: \[ 0 + 2y \leq 30 \Rightarrow y \leq 15 \] But $ x \geq 2 $, so ignore $ x=0 $. $ y=0 $: \[ x + 0 \leq 30 \Rightarrow x \leq 30 \] But $ y \geq 2 $, so ignore $ y=0 $. Determine the maximum total tickets: At point $(26, 2)$: total tickets = $ 26 + 2 = 28 $. At point $(2, 14)$: total tickets = $ 2 + 14 = 16 $. The maximum total tickets are at $(26, 2)$, with 28 tickets. Summary: The maximum number of tickets of each kind that can be given away without exceeding $300 is **26 tickets of $10 and 2 tickets of $20**. The total number of tickets given away in this case is 28. Note: The question asks for how many of each kind can be given away, so the answer is 26 tickets of $10 and 2 tickets of $20.

Home /
solutions /
Math /
A radio station is giving away tickets to a play. They plan to give away tickets to seats that cost $10 or $20. They plan to give away at least 20 tickets, and the total cost of all the tickets can be no more than $300. Make a graph showing how many tickets of each kind can be given away.

A radio station is giving away tickets to a play. They plan to give away tickets to seats that cost $10 or $20. They plan to give away at least 20 tickets, and the total cost of all the tickets can be no more than $300. Make a graph showing how many tickets of each kind can be given away.

Answer

Answer:
The maximum number of tickets of each kind that can be given away without exceeding $300 is 10 tickets of each kind.

Explanation:
This problem involves setting up and solving a system of inequalities based on the given constraints: the total cost and the minimum number of tickets of each type. The key concepts involved are linear inequalities, systems of inequalities, and graphical solution methods for linear programming problems.

Steps:

Define variables:

Let $$ x $$ = number of $10 tickets
Let $$ y $$ = number of $20 tickets

Write the constraints:

Cost constraint:

Total cost cannot exceed $300:

10x + 20y \leq 300

Minimum tickets constraint:

At least 2 tickets of each kind:

x \geq 2

y \geq 2

Simplify the cost inequality:

Divide the entire inequality by 10:

x + 2y \leq 30

Graphical interpretation:

Plot the line $$ x + 2y = 30 $$ on the coordinate plane.
The feasible region is the area below or on this line, and to the right of the lines $$ x=2 $$ and $$ y=2 $$ .

Find the maximum number of tickets:

Since the goal is to find the maximum number of tickets of each kind without exceeding $300, and considering the constraints $x \geq 2$ , $y \geq 2$ , the maximum occurs at the corner point of the feasible region.

Determine the corner points:

When $$ y=2 $$ :

x + 2(2) = 30 \Rightarrow x + 4 = 30 \Rightarrow x=26

Check if

x \geq 2

: yes, so point

\((26, 2)\)

When $$ x=2 $$ :

2 + 2y = 30 \Rightarrow 2y=28 \Rightarrow y=14

Check if

y \geq 2

: yes, so point

\((2, 14)\)

Intersection with axes (if applicable):
$$ x=0 $$ :

0 + 2y \leq 30 \Rightarrow y \leq 15

But

x \geq 2

, so ignore

\( x=0 \)

$$ y=0 $$ :

x + 0 \leq 30 \Rightarrow x \leq 30

But

y \geq 2

, so ignore

\( y=0 \)

Determine the maximum total tickets:

At point $$(26, 2)$$ : total tickets = $$ 26 + 2 = 28 $$ .
At point $$(2, 14)$$ : total tickets = $$ 2 + 14 = 16 $$ .

The maximum total tickets are at $$(26, 2)$$ , with 28 tickets.

Summary:

The maximum number of tickets of each kind that can be given away without exceeding $$300 is **26 tickets of $$ 10 and 2 tickets of $20**.
The total number of tickets given away in this case is 28.

Note: The question asks for how many of each kind can be given away, so the answer is 26 tickets of $$10 and 2 tickets of $$ 20.

Rename 4 thousands 7 hundred = 47

Michael, the art elective programme student, is working on another assignment. He designs a rectangular pattern measuring 45 mm by 42 mm. He is required to use identical rectangular patterns to form a square. The maximum area of the square allowed is 1.6m^2. (i) How many patterns does he need to form the smallest square? (ii) What are the dimensions of the largest square that he can form?

Which formula/name pair is incorrect? A) FeSO4 iron(II) sulfate B) Fe2(SO3)3 iron(III) sulfate C) FeS iron(II) sulfide D) FeSO3 iron(II) sulfide E) Fe2(SO4)3 iron(III) sulfide

Kavitha wanted to buy a laptop. She saved 1/3 of the cost of the laptop in the first month. In the second month she saved $125 less than what she saved in the first month. She saved the remaining $525 in the third month. How much did the laptop ost? enjamin bought tokens at a funfair. He used 3/8 of them at the ring-toss booth and 5 of the remaining tokens at the darts booth. He then bought another 35 tokens and d 10 tokens more than what he had at first. How many tokens did Benjamin have irst? e number of fiction books at a library was 3/4 of the total number of books. After fiction books were donated to the library, the number of fiction books became 5/6 e total number of books. How many books were there at the library at first?

Which division expression is shown in the model?

Identify the elements correctly shown by decreasing radii size. N3to N S->S2- N>N3- Cu2+>Cu+ K+>K

20.- The circle graph gives the distribution of salad dressing chosen by customers at a restaurant. If approximately 200 customers order salad each day at the restaurant, which of the following is closest to the difference per day between the number who choose Italian dressing, and the number who chose vinaigrette dressing? a) 10 Salad Dressing b) 20 Italian c) 30 Vinaigrette d) 50 Blue Cheese French

The table shows the results of tossing a chipped number cube 80 times. Players A and B decide to play a game. If the roll is even, Player A wins. Otherwise, Player B wins. Is the game fair? Explain.